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Bose-Einstein Condensate in Traps: a Diffusion Monte Carlo Analysis J. L

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Bose-Einstein Condensate in Traps: a Diffusion Monte Carlo Analysis J. L Bose-Einstein condensate in traps: A Diffusion Monte Carlo analysis J. L. DuBois, H. R. Glyde Department of Physics and Astronomy, University of Delaware Newark, Delaware 19716, USA 1. INTRODUCTION The first demonstration of Bose-Einstein condensation in gases of alkalai atoms in magnetic traps in 1995 ignited a dramatic increase in interest in finite sized Bose systems [1-3]. In the initial experiments (e.g. 87Rb in harmonic traps) the Bose gas was dilute. However, trapped Bose systems with higher densities are now being inves- tigated [4-5]. As the density or interaction strength is increased, condensate depletion increases and effects of interaction between the condensate and non-condensate be- come important. In this paper we explore the properties of trapped Bose gases over a wide range of densities using Monte Carlo (MC) methods. We evaluate the total energy, the distribution of Bosons in the trap, the condensate fraction, the distribution of the condensate throughout the trap and other properties as a function of density. We go from the dilute gas limit to densities comparable to liquid 4He densities. We compare the MC results with mean field theories (e.g. Gross-Pitaevskii (GP) theory of the condensate, the Bogoliubov expression for the condensate fraction) and determine the densities at which MC and mean field results begin to differ. We find that the condensate moves from the center of the trap in the dilute limit to the outer surface of the gas at high densities. We present particularly new Diffusion Monte Carlo (DMC) results and discuss some technical problems that arise in DMC applied to a finite sized system in a central potential. The density of Bosons in a trap and the importance of interatomic interactions is conveniently characterized by the parameter, n(0)a3 = N(0)a3=V . Here n(0) = N(0)=V is the number density at the center of the trap and a is the scattering length (or hard sphere diameter in a hard sphere model of the atom). In a uniform system, the na3 is the ratio of the volume occupied by the Bosons, Na3, to the total volume (V ) of the gas. At a ! 0, the gas is ideal. In the dilute limit (e.g. 87Rb in a trap) where n(0)a3 ¼ 10¡5, the interaction between Bosons is important. However, it can Figure 1. Range of system densities considered in this work expressed in terms of na3 ´ Na3=V – The ratio of the volume occupied by N hard core particles with diameter a to the total volume of the system, V . be reliably treated using mean field theories (e.g GP theory). At these densities, depletion of the condensate by the interaction is negligible (· 1%). Liquid 4He at SVP, na3 = 0:21, is, in contrast, a strongly interacting liquid requiring full treatment of correlations induced by the interaction. The range of densities of trapped Bose systems investigated to date is displayed in Fig. 1. Of special interest is 85Rb for which the scattering length, a, can be adjusted to large positive and negative values by using a Feshbach resonance [4-5]. With this technique trapped Bosons of variable density can be created. To sketch the physical properties of trapped Bosons, we note that the energy of 87 a single particle in a typical harmonic trap (e.g. Rb) is (3=2)¯h!ho ¼ 10 nK. Since the mean square vibrational amplitude of a particle in a trap with trap frequency 2 !ho is hu i = (¯h=2m!ho), a useful measure of the gas size or “trap length” is aho ´ 1=2 87 (¯h=m!ho) . The scattering length of Rb is a ¼ 50A˚ ¼ 100a0, where a0 = 0:529A˚ is the Bohr radius. In this work we will sometimes use a ratio of scattering length to trap length which corresponds to the initial experiments of Anderson et. al [1] ¡3 (aRb=aho = 4:33 £ 10 ) as a benchmark. Introducing the mean distance between ¡3 Bosons, n = N=V =r ¯ , we have, a ¿ r¯ ¿ aho in the dilute limit and a ¼ r¯ ¿ aho in the dense limit. In the dilute limit, a ¿ r¯, the interatomic interaction is weak and the trap potential dominates. In the dense limit a ¼ r¯, the interatomic interaction dominates, the trap potential is negligible and the gas exhibits properties comparable to a self bound liquid (e.g. a liquid 4He droplet). To achieve BEC, the thermal de 2 1=2 3> Broglie wavelength ¸T = (h =2¼mkT ) must be large enough so that n¸ » 2:616. 3 That is, ¸R ¼ r¯. The critical temperature of BEC determined by n¸T c = 2:616 is 4 3 3 ¡4 Tc ¼ 75 £ 10 (na ) nK. For na = 10 and T = 0:1 Tc, a temperature T = 7:5 nK similar to the trap energy is required. We consider only T = 0K here. 2. Bosons in Traps: Monte Carlo Formulation We consider N Bosons of mass m confined in an external trapping potential, Vext(r), and interacting via a two-body potential, Vint(r1; r2). The Hamiltonian for this system is: µ ¶ XN ¡¯h2 XN H = r2 + V (r ) + V (r ; r ); (1) 2m i ext i int i j i i<j where the trapping potential is spherically symmetric and harmonic, 1 V (r) = m!2 r2: (2) ext 2 ho 2 Here !ho defines the trap potential strength. We have chosen to represent the inter- Boson interaction by a pairwise, hard sphere potential with diameter a. Vint(r) is zero if the Bosons are separated by a distance r greater than a but infinite if they attempt to come within a distance r · a. In the low energy limit, the scattering length between a pair of particles interacting via a hard core potential is purely s- wave with scattering length a. Similarly, in this limit the scattering between a pair of particles interacting via the contact potential 4¼¯h2a v(r) = g±(r) = ±(r) (3) m is also purely s-wave with scattering length a. Thus properties of H defined in (1-2) and evaluated using MC can be compared directly with properties calculated using (3) and the GP equation in the limit where the s-wave approximation is valid. Systems of N = 128 to N = 1024 particles with varying values of a are considered. We express lengths in units of the trap length, aho, and energies in units of the trap energy, ¯h!ho, introduced above. As a trial variational wave-function we use YN YN ΨT = Á(ri) f(jri ¡ rjj): (4) i=1 i<j The single Bosons component with variational parameters, ®0, and ®1 is 2 4 Á(r) = e¡(®0r +®1r ): (5) We used two pair Jastrow functions, f0(rij ) = (1 ¡ a=rij ) (6) denoted VMC0 and 2 f1(rij) = (1 ¡ a=rij )exp[¯0exp((rij ¡ a) =¯1)] (7) denoted VMC1. The f0(rij) is the exact solution for a pair of particles at low energy interacting via a hard core potential. The f1(rij) contains a correction term which improves this and has two variational parameters, ¯0 and ¯1. The variational MC energy is, using (4), EVMC = hΨT jHjΨT i=hΨT jΨT i. EDMC is the usual Diffusion MC energy obtained using ΨT as the initial and guiding function [6]. We wish to compare MC energies with mean field energies. For example, the energy of a uniform hard core Bose gas (Vext = 0) at T = 0 K is [7], ³ 2 ´h i 3 ¯h 128 3 1 E=N = 4¼na 1 + p (na ) 2 + ::: : (8) 2ma2 15 ¼ Here 4¼na3 is the energy per Boson of the condensate (in units of ¯h2=2ma2). The term in (na3) represents the increase in E=N arising from Bosons excited out of (or depleted from) the condensate by interactions. This “depletion” term can also be incorporated into the GP energy functional’s interaction term in a local density approximation. The resulting modified Gross-Pitaevskii (MGP) equation gives an E=N for Bosons in a trap in the large Na (Thomas-Fermi) limit of 5 h 7 i E =N = ¹ 1 + (¼n (0)a3)1=2 (9) MGP 7 TF 8 TF where ¹TF = gnTF (0) [7]. Here nTF (0) is the number density of the Bosons at the center of the trap in the Thomas Fermi limit. The difference between EMGP and EGP = 5=7¹TF arising from depletion is, 3=5 35 (15) 3=5 a 8=5 (EMGP ¡ EGP )=N¯h!ho = p N ( ) (10) 112 8 aho Particularly we note that the increase in E arising from depletion is proportional to 3=5 8=5 N (a=aho) . Condensate properties are obtained by diagonalizing the one-body density ma- trix (OBDM) as described in ref [8]. The OBDM is [9] ½(r0; r) = hΨˆ y(r0); Ψ(ˆ r)i; (11) where Ψ(ˆ r) is the field operator that annihilates a single particle at the point r in the system. The OBDM, ½(r0; r), characterizes the correlations which exist between the points r and r0 in a many-body state. We consider spherically symmetric traps only. For a spherically symmetric system, the OBDM may be expanded in terms of the Legendre polynomials, Pl, as X (2l + 1) ½(r; r0) = P (ˆr ¢ ˆr0)½ (r; r0): (12) 4¼ l l l 0 Here, ½l (r1 ; r1 ) may be approximated in terms of the optimized trial wave-function, Ψº , as Z 0 ¤ 0 0 ½l (r1 ; r1 ) = dΩ1dr2 ::drN Ψº (r1 ::rN )Pl(ˆr1 ¢ rˆ1 )Ψº (r1 ::rN ): (13) 0 Each ½l (r1 ; r1 ) is a matrix and may be diagonalized to obtain the orbitals Áln(r) and their corresponding occupation numbers nln.
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